While browsing the plug-in code I've found the filter 'frm_fields_for_csv_export' (https://formidableforms.com/knowledgebase/frm_fields_for_csv_export/) which actually is being used before attaching the CSV to an email.  This is the code I eventually used to make it work.

add_filter( 'frm_fields_for_csv_export', 'sv_change_headers_for_csv_export', 10, 2 ); 
function sv_change_headers_for_csv_export( $fields, $args ) {  
    $target_form_id = 3; // change 220 to the ID of the form  
    if ( $target_form_id === (int) $args['form']->id ) {   
        // change 67 to your field ids, and the text to the new CSV headers   
        $matches = array(    
            67  => 'New name'   
        );   
        // Loops all $fields and changes the names of the fields that are exported to CSV   
        foreach ( $fields as $field ) {    
            if ( isset($matches[$field->id] ) ) {     
                $field->name = $matches[$field->id];    
            }   
        }  
    }  
    return $fields; 
}